Selected Topics on Hermite-Hadamard Inequalities and Applications 

Contents

1. Preface v
Chapter 1.  Introduction  1.
1. Historical Considerations  1
2. Characterisations of Convexity via H.-H. Inequalities  3
3. Some Generalisations  5
Chapter 2. Some Results Related to the H.-H. Inequality  9
1. Generalisations of the H.-H. Inequality  9
2. Hadamard’s Inferior and Superior Sums  20
3. A Refinement of the H.-H. Inequality for Modulus  26
4. Further Inequalities for Differentiable Convex Functions  29
5. Further Inequalities for Twice Differentiable Convex Functions  39
6. A Best Possible H.-H. Inequality in Fink’s Sense  55
7. Generalised Weighted Mean Values of Convex Functions  61
8. Generalisations for n - Time Differentiable Functions  66
9. The Euler Formulae and Convex functions  70
10. H.-H. Inequality for Isotonic Linear Functionals  79
11. H.-H. Inequality for Isotonic Sublinear Functionals  85
Chapter 3.  Some Functionals Associated with the H.-H. Inequality  93
1. Two Difference Mappings  93
2. Properties of Superadditivity and Supermultiplicity  100
3. Properties of Some Mappings Defined By Integrals  108
4. Some Results due to B.G. Pachpatte  129
5. Fejer’s Generalization of the H.-H. Inequality  132
6. Further Results Refining the H.-H. Inequality  142
7. Another Generalisation of Fejer’s Result  150
Chapter 4.  Sequences of Mappings Associated with the H.-H. Inequality  157
1. Some Sequences Defined by Multiple Integrals  157
2. Convergence Results  163
3. Estimation of Some Sequences of Multiple Integrals  168
4. Further Generalizations  178
5. Properties of the Sequence of Mappings Hn  182
6. Applications for Special Means  199
Chapter 5. The H.-H. Inequality for Different Kinds of Convexity  201
1. Integral Inequalities of H.-H. Type for Log-Convex Functions  201
2. The H.-H. Inequality for r - Convex Functions  209
3. Stolarsky Means and H.-H.’s Inequality  215
4. Functional Stolarsky Means and H.-H. Inequality  222
5. Generalization of H.-H. Inequality for G-Convex Functions  230
6. H.-H. Inequality for the Godnova-Levin Class of Functions  240
7. The H.-H. Inequality for Quasi-Convex Functions  248
8. P - functions, Quasiconvex Functions and H.-H. Type Inequalities  254
9. Convexity According to the Geometric Mean  262
10. The H.-H. Inequality of s - Convex Functions in the First Sense  281
11. The Case for s - Convex Functions in the Second Sense  290
12. Inequalities for m - Convex and (a; m) • Convex Functions  297
13. Inequalities for Convex-Dominated Functions  304
14. H.-H. Inequality for Lipschitzian Mappings  311
Chapter 6. The H.-H. Inequalities for Mappings of Several Variables  319
1. An Inequality for Convex Functions on the Co-ordinates  319
2. A H.-H. Inequality on the Disk  327
3. A H.-H. Inequality on a Ball  336
4. A H.-H. Inequality for Functions on a Convex Domain  344
Bibliography  349